What is the energy required to launch a `m` kg satellite from earth's surface in a circular orbit at an altitude of 7R ? (R = radius of the earth)

Options

A. `(2)/(3) mgR`
B. `mgR`
C. `(5)/(6)mgR`
D. `(1)/(3)mgR`

Explanation

To find the energy required to launch a satellite of mass \( m \) kg from the Earth's surface to a circular orbit at an altitude of \( 7R \) (where \( R \) is the radius of the Earth), we can follow these steps:

### Step 1: Determine the radius of the orbit
The altitude of the satellite is given as \( 7R \). Therefore, the total distance from the center of the Earth to the satellite (the orbital radius \( r \)) is:
\[ r = R + 7R = 8R \]

### Step 2: Calculate the gravitational potential energy at the Earth's surface
The gravitational potential energy (U) at the Earth's surface is given by:
\[ U_{\text{surface}} = -\frac{GMm}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the satellite.

### Step 3: Calculate the gravitational potential energy in the orbit
The gravitational potential energy at the altitude of \( 7R \) (or at a distance \( 8R \) from the center of the Earth) is:
\[ U_{\text{orbit}} = -\frac{GMm}{8R} \]

### Step 4: Calculate the total mechanical energy in the orbit
The total mechanical energy (E) in a circular orbit is given by:
\[ E = K + U \]
where \( K \) is the kinetic energy. The total energy in the orbit can also be expressed as:
\[ E = -\frac{GMm}{2r} = -\frac{GMm}{2 \times 8R} = -\frac{GMm}{16R} \]

### Step 5: Calculate the energy required to launch the satellite
The energy required to launch the satellite is the difference in potential energy from the Earth's surface to the orbit:
\[ \Delta E = U_{\text{orbit}} - U_{\text{surface}} \]
Substituting the values from Steps 2 and 3:
\[ \Delta E = \left(-\frac{GMm}{8R}\right) - \left(-\frac{GMm}{R}\right) \]
\[ \Delta E = -\frac{GMm}{8R} + \frac{GMm}{R} \]
\[ \Delta E = \frac{GMm}{R} - \frac{GMm}{8R} = \frac{GMm}{R} \left(1 - \frac{1}{8}\right) = \frac{GMm}{R} \cdot \frac{7}{8} \]

### Step 6: Substitute \( g = \frac{GM}{R^2} \)
We know that \( g = \frac{GM}{R^2} \), so we can express \( GM \) as \( gR^2 \):
\[ \Delta E = \frac{gR^2 m}{R} \cdot \frac{7}{8} = \frac{7}{8} gR m \]

### Final Answer
Thus, the energy required to launch the satellite is:
\[ \Delta E = \frac{7}{8} g R m \]